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Theorem oppcinv 16372
Description: An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
oppcsect.b 𝐵 = (Base‘𝐶)
oppcsect.o 𝑂 = (oppCat‘𝐶)
oppcsect.c (𝜑𝐶 ∈ Cat)
oppcsect.x (𝜑𝑋𝐵)
oppcsect.y (𝜑𝑌𝐵)
oppcinv.s 𝐼 = (Inv‘𝐶)
oppcinv.t 𝐽 = (Inv‘𝑂)
Assertion
Ref Expression
oppcinv (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

Proof of Theorem oppcinv
StepHypRef Expression
1 incom 3788 . . 3 ((𝑋(Sect‘𝑂)𝑌) ∩ (𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌))
2 oppcsect.b . . . . . . 7 𝐵 = (Base‘𝐶)
3 oppcsect.o . . . . . . 7 𝑂 = (oppCat‘𝐶)
4 oppcsect.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
5 oppcsect.y . . . . . . 7 (𝜑𝑌𝐵)
6 oppcsect.x . . . . . . 7 (𝜑𝑋𝐵)
7 eqid 2621 . . . . . . 7 (Sect‘𝐶) = (Sect‘𝐶)
8 eqid 2621 . . . . . . 7 (Sect‘𝑂) = (Sect‘𝑂)
92, 3, 4, 5, 6, 7, 8oppcsect2 16371 . . . . . 6 (𝜑 → (𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
109cnveqd 5263 . . . . 5 (𝜑(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
11 eqid 2621 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
12 eqid 2621 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
13 eqid 2621 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
142, 11, 12, 13, 7, 4, 5, 6sectss 16344 . . . . . . 7 (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
15 relxp 5193 . . . . . . 7 Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
16 relss 5172 . . . . . . 7 ((𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌(Sect‘𝐶)𝑋)))
1714, 15, 16mpisyl 21 . . . . . 6 (𝜑 → Rel (𝑌(Sect‘𝐶)𝑋))
18 dfrel2 5547 . . . . . 6 (Rel (𝑌(Sect‘𝐶)𝑋) ↔ (𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
1917, 18sylib 208 . . . . 5 (𝜑(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
2010, 19eqtrd 2655 . . . 4 (𝜑(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
212, 3, 4, 6, 5, 7, 8oppcsect2 16371 . . . 4 (𝜑 → (𝑋(Sect‘𝑂)𝑌) = (𝑋(Sect‘𝐶)𝑌))
2220, 21ineq12d 3798 . . 3 (𝜑 → ((𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ (𝑋(Sect‘𝐶)𝑌)))
231, 22syl5eq 2667 . 2 (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ (𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ (𝑋(Sect‘𝐶)𝑌)))
243, 2oppcbas 16310 . . 3 𝐵 = (Base‘𝑂)
25 oppcinv.t . . 3 𝐽 = (Inv‘𝑂)
263oppccat 16314 . . . 4 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
274, 26syl 17 . . 3 (𝜑𝑂 ∈ Cat)
2824, 25, 27, 6, 5, 8invfval 16351 . 2 (𝜑 → (𝑋𝐽𝑌) = ((𝑋(Sect‘𝑂)𝑌) ∩ (𝑌(Sect‘𝑂)𝑋)))
29 oppcinv.s . . 3 𝐼 = (Inv‘𝐶)
302, 29, 4, 5, 6, 7invfval 16351 . 2 (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ (𝑋(Sect‘𝐶)𝑌)))
3123, 28, 303eqtr4d 2665 1 (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cin 3558  wss 3559   × cxp 5077  ccnv 5078  Rel wrel 5084  cfv 5852  (class class class)co 6610  Basecbs 15792  Hom chom 15884  compcco 15885  Catccat 16257  Idccid 16258  oppCatcoppc 16303  Sectcsect 16336  Invcinv 16337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-hom 15898  df-cco 15899  df-cat 16261  df-cid 16262  df-oppc 16304  df-sect 16339  df-inv 16340
This theorem is referenced by:  oppciso  16373  episect  16377
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